Page 46 - Chemical engineering design

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INTRODUCTION TO DESIGN
likened to seeking the top of a hill (or bottom of a valley), and a useful technique for
this type of problem is the gradient method (method of steepest ascent, or descent), see
Edgar and Himmelblau (2001).
1.10.4. Linear programming
Linear programming is an optimisation technique that can be used when the objective
function and constraints can be expressed as a linear function of the variables; see Driebeek
(1969), Williams (1967) and Dano (1965).
The technique is useful where the problem is to decide the optimum utilisation of
resources. Many oil companies use linear programming to determine the optimum schedule
of products to be produced from the crude oils available. Algorithms have been developed
for the efficient solution of linear programming problems and the SIMPLEX algorithm,
Dantzig (1963), is the most commonly used.
Examples of the application of linear programming in chemical process plant design
and operation are given by Allen (1971), Rudd and Watson (1968), Stoecker (1991), and
Urbaniec (1986).

1.10.5. Dynamic programming
Dynamic programming is a technique developed for the optimisation of large systems;
see Nemhauser (1966), Bellman (1957) and Aris (1963).
The basic approach used is to divide the system into convenient sub-systems and
optimise each sub-system separately, while taking into account the interactions between
the sub-systems. The decisions made at each stage contribute to the overall systems
objective function, and to optimise the overall objective function an appropriate combi-
nation of the individual stages has to be found. In a typical process plant system the
possible number of combinations of the stage decisions will be very large. The dynamic
programming approach uses Bellman’s “Principle of Optimality”, † which enables the
optimum policy to be found systematically and efficiently by calculating only a fraction
of the possible combinations of stage decisions. The method converts the problem from
the need to deal with “N” optimisation decisions simultaneously to a sequential set of “N”
problems. The application of dynamic programming to design problems is well illustrated
in Rudd and Watson’s book; see also Wells (1973) and Edgar and Himmelblau (2001).

1.10.6. Optimisation of batch and semicontinuous processes
In batch operation there will be periods when product is being produced, followed by non-
productive periods when the product is discharged and the equipment prepared for the
next batch. The rate of production will be determined by the total batch time, productive

† Bellman’s (1957) principle of optimality: “An optimal policy has the property that, whatever the initial state
and the initial decision are, the remaining decisions must constitute an optimal policy with regard to the state
resulting from the first decision.”